In my opinion, this a choice between two-part epoxy and polyurethane glue (eg, Gorilla Glue). Both are messy, but both would hold much better than CA, and both offer enough 'working time' to get the spoon fully embedded in the handle. The difference between them is the additional step of having to mix epoxy, versus dealing with the foam-out if you use polyurethane.
In your case, the shaft of the spoon is a spiral, and that's a very good thing for two reasons. First, it means that when whatever glue you use cures, it will harden into the shape of that spiral. So the shaft will effectively be screwed and glued into the handle, which means that it will not easily come out. You will want to use enough glue to fill the flutes of the spiral, and one of the challenges is that there is no easy way to know how much glue is enough. Using a glue that gives you more working time provides flexibility if you need to add more glue. You will want to get some glue into the hole before inserting the shaft - that glue will be forced deeper into the hole to secure the bond, but there will likely be some excess glue that will be forced back out of the hole around the shaft that you will have to clean up.
The worst case for this kind of situation would be where the shaft is smooth - in that case, inserting the shaft is just like putting a cork in a bottle - it seals the cavity, trapping the air in the hole under that seal. So forcing the shaft into the hole compress the air which means that when you let go, it will try to expand and force the shaft back out of the hole. There are two ways to deal with that problem - one is to use a clamp to secure the spoon in the handle until the glue cures. There usually is no problem having a pocket of compressed air at the bottom of the hole (and over time that air will work its way out through the fibers of the wood in the handle). Another option is to provide a channel for the compressed air to escape. You could use a Dremel to cut a shallow kerf along the length of the shaft, or you could drill a small 'breather hole' in the wooden handle near the bottom of the hole. In either case, expect to have some glue come out of the the channel as you press the shaft into the hole.